Linear and Integer Programming :Algorithms in the Real World. Related Optimization Problems. How important is optimization?

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1 Linear and Integer Programming :Algorithms in the Real World Linear and Integer Programming I Introduction Geometric Interpretation Simplex Method Linear or Integer programming maximize z = c T x cost or objective function subject to Ax = b equalities x 0 inequalities c R n, b R m, A R m n Linear programming: x R n (polynomial time) Integer programming: x Z n (NP-complete) Extremely general framework, especially IP Page Page2 Related Optimization Problems Unconstrained optimization max{f(x) : x R n } Constrained optimization max{f(x) : g i (x) 0, h j (x) = 0} Quadratic programming max{1/2x T Qx + c T x : Ax b, Ex = d} Zero-One programming max{c T x : Ax = b, x {0,1} n, c R n, b R m } Mixed Integer Programming max{c T x : Ax = b, x 0, x i Z n, i I, x r R n, r R} How important is optimization? 50+ packages available papers just on interior-point methods 100+ books in the library 10+ courses at most Universities 100s of companies All major airlines, delivery companies, trucking companies, manufacturers, make serious use of optimization Page Page4 1

2 Linear+Integer Programming Outline Linear Programming General formulation and geometric interpretation Simplex method Ellipsoid method Interior point methods Integer Programming Various reductions of NP hard problems Linear programming approximations Branch-and-bound + cutting-plane techniques Applications of Linear Programming 1. A substep in most integer and mixed-integer linear programming (MIP) methods 2. Selecting a mix: oil mixtures, portfolio selection 3. Distribution: how much of a commodity should be distributed to different locations. 4. Allocation: how much of a resource should be allocated to different tasks 5. Network Flows Page Page6 Linear Programming for Max-Flow in 5 3 Create two variables per edge: Create one equality per vertex: + + = + + and two inequalities per edge: 3, 3 add edge x 0 from out to in maximize x out Page7 In Practice In the real world most problems involve at least some integral constraints. Many resources are integral Can be used to model yes/no decisions (0-1 variables) Therefore 1. A subset in integer or MIP programming is the most common use in practice Page8 2

3 Algorithms for Linear Programming Simplex (Dantzig 1947) Ellipsoid (Kachian 1979) first algorithm known to be polynomial time Interior Point first practical polynomial-time algorithms Projective method (Karmakar 1984) Affine Method (Dikin 1967) Log-Barrier Methods (Frisch 1977, Fiacco 1968, Gill et.al. 1986) Many of the interior point methods can be applied to nonlinear programs. Not known to be poly. time State of the art 1 million variables 10 million nonzeros No clear winner between Simplex and Interior Point Depends on the problem Interior point methods are subsuming more and more cases All major packages supply both The truth: the sparse matrix routines, make or break both methods. The best packages are highly sophisticated Page Page10 problem Comparisons, 1994 Simplex (primal) Simplex (dual) Barrier + crossover binpacking distribution 18,568.0 won t run too big forestry 1, , ,348.0 maintenace 57, , ,240.8 crew 7, , ,264.2 airfleet 71, , ,627.3 energy 3, , color 45,870.2 won t run too big Formulations There are many ways to formulate linear programs: objective (or cost) function maximize c T x, or minimize c T x, or find any feasible solution (in)equalities Ax b, or Ax b, or Ax = b, or any combination nonnegative variables x 0, or not Fortunately it is pretty easy to convert among forms Page Page12 3

4 Formulations The two most common formulations: Canonical form maximize c T x subject to Ax b x 0 e.g , 0 slack variables y 1 More on slack variables later. Standard form maximize c T x subject to Ax = b x y 1 = 7,, y Page13 Geometric View A polytope in n-dimensional space Each inequality corresponds to a half-space. The feasible set is the intersection of the halfspaces This corresponds to a polytope Polytopes are convex: if x,y is in the polytope, so is the line segment joining them. The optimal solution is at a vertex (i.e., a corner). Simplex moves around on the surface of the polytope Interior-Point methods move within the polytope Page14 Geometric View Notes about higher dimensions maximize: z = subject to: , 0 An intersection of 5 halfspaces Feasible Set Objective Function Corners Page15 For n dimensions and no degeneracy Each corner (extreme point) consists of: n intersecting (n-1)-dimensional hyperplanes e.g. for n = 3, 3 intersecting 2d planes make corner n intersecting edges Each edge corresponds to moving off of one hyperplane (still constrained by n-1 of them) # Corners can be exponential in n (e.g., a hypercube) Simplex will move from corner to corner along the edges Page16 4

5 The Simple Essense of Simplex Polytope P Input: max f(x) = cx s.t. x in P = {x : Ax b, x 0} Consider Polytope P from canonical form as a graph G = (V,E) with V = polytope vertices, E = polytope edges. 1) Find any vertex v of P. 2) While there exists a neighbor u of v in G with f(u) < f(v), update v to u. 3) Output v. Choice of neighbor if several u have f(u) < f(v)? Optimality and Reduced Cost The Optimal solution must include a corner. The Reduced cost for a hyperplane at a corner is the cost of moving one unit away from the plane along its corresponding edge. z 1 p i e i r i = -z e i For maximization, if all reduced cost are nonnegative, then we are at an optimal solution. Finding the most negative reduced cost is one often used heuristic for choosing an edge to leave on Page Page18 Reduced cost example Simplex Algorithm z = e i e i = (2, 1) Ex: reduced cost for leaving -axis from point (4,0) Moving 1 unit off of - axis will move us (2,1) units along the edge. The reduced cost of leaving the plane is -(2,3) (2,1) = Page19 1. Find a corner of the feasible region 2. Repeat A. For each of the n hyperplanes intersecting at the corner, calculate its reduced cost B. If they are all non-negative, then done C. Else, pick the most negative reduced cost This is called the entering plane D. Move along corresponding edge (i.e. leave that hyperplane) until we reach the next corner (i.e. reach another hyperplane) The new plane is called the departing plane Page20 5

6 Example Simplifying Departing Step 1 Entering Step 2 z = Start Problem: The Ax b constraints not symmetric with the x 0 constraints. We would like more symmetry. Idea: Leave only inequalities of the form x 0. Use slack variables to do this. Convert into form: maximize c T x subject to Ax = b x Page Page22 maximize c T x subject to Ax b x 0 A = m n i.e. m equations, n variables Standard Form slack variables Standard Form maximize c T x subject to A x = b x 0 A = m (m+n) i.e. m equations, m+n variables = Page23 maximize: z = subject to: 2 + = = 18 + = 10,,,, 0 Example, again The equality constraints impose a 2d plane embedded in 5d space, looking at the plane gives the figure above Page24 6

7 Using Matrices If before adding the slack variables A has size m n then after it has size m (n + m) m can be larger or smaller than n A = n m slack vrs. Assuming rows are independent, the solution space of Ax = b is an n-dimensional subspace. m Page25 Simplex Algorithm, again 1. Find a corner of the feasible region 2. Repeat A. For each of the n hyperplanes intersecting at the corner, calculate its reduced cost B. If they are all non-negative, then done C. Else, pick the most negative reduced cost This is called the entering plane D. Move along corresponding line (i.e. leave that hyperplane) until we reach the next corner (i.e. reach another hyperplane) The new plane is called the departing plane Page26 Simplex Algorithm (Tableau Method) m reduced costs n F r Free Variables I b 0 z Basic Vars. current cost This form is called a Basic Solution the n free variables are set to 0 the m basic variables are set to b A valid solution to Ax = b if reached using Gaussian Elimination Represents n intersecting hyperplanes If feasible (i.e. b 0), then the solution is called a Basic Feasible Solution and is a corner of the feasible set Corner = = 18 + = z = 0 Free variables Basic Vars Page Page28 7

8 Corner Corner free variables basic variables Page Page30 Corner Corner Page Page32 8

9 Corner Simplex Method Again Note that in general there are n+m choose m corners Once you have found a basic feasible solution (a corner), we can move from corner to corner by swapping columns and eliminating. ALGORITHM 1. Find a basic feasible solution 2. Repeat A. If r (reduced cost ) 0, DONE B. Else, pick column with most negative r C. Pick row with least positive b /(selected column) D. Swap columns E. Use Gaussian elimination to restore form Page Page34 Tableau Method Tableau Method A. If r are all non-negative then done n B. Else, pick the most negative reduced cost This is called the entering plane F I b n r Free Variables values are 0 0 Basic Variables z current cost F r I 0 b z reduced costs if all 0 then done min{r i } entering variable Page Page36 9

10 Tableau Method C. Move along corresponding line (i.e. leave that hyperplane) until we reach the next corner (i.e. reach another hyperplane) The new plane is called the departing plane F r u I 0 1 b z min positive b j /u j departing variable Tableau Method D. Swap columns r swap E. Gauss-Jordan elimination F i+1 x x x b x x z I b i+1 r i+1 0 z i+1 No longer in proper form Back to proper form Page Page38 Example = = 18 + = z = 0 Find corner = = 0 (start) Page39 Example b j /v j min positive x Page40 10

11 swap Example 18 Example Gauss-Jordan Elimination Page Page42 swap Example 18 Simplex Concluding remarks For dense matrices, takes O(n(n+m)) time per iteration Can take an exponential number of iterations. In practice, sparse methods are used for the iterations Gauss-Jordan Elimination Page Page44 11

12 Duality Primal (P): maximize z = c T x subject to Ax b x 0 (n equations, m variables) Dual (D): minimize z = y T b subject to A t y c y 0 (m equations, n variables) Duality Theorem: if x is feasible for P and y is feasible for D, then cx yb and at optimality cx = yb. feasible solutions for Primal (maximization) Duality (cont.) Optimal solution for both feasible solutions for Dual (minimization) Quite similar to duality of Maximum Flow and Minimum Cut. Useful in many situations Page Page46 Duality Example Primal: maximize: z = subject to: , 0 Dual: minimize: z = 4y y y 3 subject to: y 1 + 2y 2 2-2y 1 + Y 2 + Y 3 3 y 1, y 2, y 3 0 Solution to both is 38 ( =4, =10), (y 1 =0, y 2 =1, y 3 =2) Page47 12